Placement of abrasive particles for achieving orientation independent scratches and minimizing observable manufacturing defects

ABSTRACT

The present invention includes an abrasive tool comprising a substrate and plurality of abrasive grains arranged in a pseudo-random pattern on the substrate. The abrasive grains cover in the range of 10% to 30% of the substrate surface, and in some instances, the arrangement of abrasive grains demonstrate improved orientation independence and homogeneity in distribution of abrasive grains.

FIELD OF THE INVENTION

The present invention relates to the placement of abrasive features onan abrasive work tool.

BACKGROUND

Tools with abrasive features can be used for a variety of applications,including grinding, polishing, abrading, finishing, scouring andscrubbing. Abrasive tools can be made with a variety of substratesincluding paper, woven fabrics, nonwoven fabrics, calendared nonwovenfabrics, polymeric films, stitch-bonded fabrics, open cell foams, closedcell foams, and combinations thereof. Abrasive tools can also include avariety of types of abrasive features or abrasive grains, includingsingle abrasive grits, cutting points, and composites comprising aplurality of abrasive grits, and combinations thereof.

Abrasive features or grains can be arranged on a variety of surfaces ina variety of methods and arrangements. For example, some abrasivefeatures are randomly distributed on a surface of a work tool and placedusing coating, spraying, printing and solvent methods. Some abrasivefeatures can be arranged in a pattern on a surface of a work tool.

The arrangement or placement of abrasive features can impact manyaspects of the abrasive tool including the efficiency of cutting, theorientation in which the abrasive tool should be used, the scratchpattern left behind by the tool, and the manufacturing requirements forthe abrasive tool. Improvements in abrasive tools to allow orientationindependence, reduced visibility of scratch patterns, and ease ofmanufacturing are desired.

SUMMARY

The present disclosure provides several advantages over the state of theart. For example, the present disclosure provides for an abrasive toolthat can perform more consistently in any mode of use, or when held atany orientation relative to a cut direction. The present disclosure alsoprovides for an abrasive tool with abrasive grains arranged in a mannerto reduce visibility of scratch patterns left on a work piece. Thepresent disclosure further provides for placement of abrasive grains onan abrasive tool in a pseudo-random pattern such that slightmanufacturing defects, such as missing or extra abrasive grains, are notnoticeable to a user of the abrasive tool. The present disclosure alsoprovides for an arrangement of abrasive features that provides for moreeven distribution of mass removal from a work piece. In some instances,the present disclosure allows for an arrangement of abrasive particleson a substrate in a way that can be visually pleasing independent of theorientation or shape of the substrate.

In one aspect, the present invention includes an abrasive toolcomprising a substrate and plurality of abrasive grains arranged in apseudo-random pattern on the substrate. The abrasive grains cover in therange of 10% to 30% of the substrate surface, and the arrangement ofabrasive grains has a score of 20% or less on the OrientationIndependence Test.

In another aspect, the present invention includes an abrasive toolcomprising a substrate and a plurality of abrasive grains arranged in apseudo-random pattern on the substrate, wherein the abrasive grainscover in the range of 10% to 30% of the substrate surface and whereinthe arrangement of abrasive grains has a score in the range of 0.7 to0.9 in the Local Homogeneity Index Test.

In some instances, the arrangement of abrasive grains has a score of 10%or less on the Orientation Independent Test.

In some instances, the abrasive grains cover 10% to 15% of the surfacesubstrate.

In some instances, the substrate comprises a lofty non-woven material.

In some instances, the lofty non-woven material comprises a densifiedsurface.

In some instances, the abrasive grains are at least one of: singleabrasive grits, cutting points, and composites comprising a plurality ofabrasive grits, and combinations thereof.

In some instances, the composites comprise a plurality of abrasive gritsin a resin.

In some instances, the abrasive grains are printed onto the substrate.

In some instances, the average abrasive grain height from the surface ofthe substrate is in the range of 0.25 mm to 1.5 mm.

In some instances, the pseudo-random pattern comprises clusters ofabrasive grains.

In some instances, the substrate is selected from the group consistingof paper, woven fabrics, nonwoven fabrics, calendared nonwoven fabrics,polymeric films, stitchbonded fabrics, open cell foams, closed cellfoams, and combinations thereof.

In some instances, the substrate comprises an open cell foam or a closedcell foam laminated to a substrate selected from the group consisting ofpaper, woven fabrics, nonwoven fabrics, calendared nonwoven fabrics,polymeric films, stitchbonded fabrics, open cell foams, closed cellfoams, and combinations thereof.

In some instances, the pseudo-random pattern is a pseudo-poissonpattern.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may be more completely understood in consideration of thefollowing detailed description of various aspects of the invention inconnection with the accompanying drawings, in which:

FIG. 1 shows an example of an abrasive tool with a regular hexagonalpattern of abrasive grains.

FIG. 2 shows an image of a scratch pattern formed by the prior artabrasive tool of FIG. 1 in two different cutting directions.

FIG. 3 shows an example of a pseudo random pattern of abrasive grains,wherein the pseudo random pattern is a pseudo-poisson pattern with a 14%abrasive grain coverage.

FIG. 4 shows an example of five circular active areas on an abrasivetool with a regular hexagonal pattern of abrasive grains.

FIG. 5 shows cutting direction for an abrasive tool.

FIG. 6 shows a flow chart illustrating how to convert an abrasivepattern into data that can be used for the Orientation Independence Testand the Local Homogeneity Test.

FIG. 7 shows the results of the Orientation Independence Test forabrasive grain pattern in FIG. 4.

FIG. 8 shows the output of the Local Homogeneity Test for the abrasivegrain pattern in FIG. 4.

FIG. 9 shows the schematic for the regular hexagonal abrasive patternsused for Comparative Examples 1 and 2.

FIGS. 10a-10h show images of the patterns used for the Examples.

It is understood that the aspects of the invention may be utilized andstructural changes may be made without departing from the scope of theinvention. The figures are not necessarily to scale. Like numbers usedin the figures refer to like components. However, it is understood thatthe use of a number to refer to a component in a given figure is notintended to limit the component in another figure labeled with the samenumber.

DETAILED DESCRIPTION

FIG. 1 shows a perspective view of example of an abrasive tool 10 with aregular hexagonal pattern of abrasive grains 12. Abrasive grains 12comprise abrasive particles suspended in a resin and printed on adensified surface 16 of nonwoven web 14.

FIG. 2 shows an image of scratch patterns formed by the abrasive tool ofFIG. 1. When the abrasive tool 20 is held at the orientation shown inFIG. 2, and the tool is moved in a lateral direction as shown by arrow28, the cut patterns 22 created on a stainless steel surface. Ingeneral, a cutting surface or a work piece can be any surface lesshardness than the abrasive features.

The cut patterns 22 on surface 21 are approximately evenly spaced apartand quite distinct and visible. In this scenario, the patterns areparticularly visible because each of abrasive grains 23 a, 23 b, 23 c,and so on are all cutting along the same line, thus resulting in a verynoticeable cut or scratch pattern. When abrasive tool 24 is held at aslightly angled orientation, and the tool is moved in a lateraldirection as indicated by arrow 28, the resulting cut or scratch lines26 are more randomly spaced apart, but are still quite visible

Further, if a single abrasive grain was missed during the manufacturingprocess of abrasive tools 20 and 24, or if an extra abrasive grain wasadded to one of the pads, it would be visually apparent to even a casualobserver. This results in a more challenging manufacturing qualitystandard and greater waste from the manufacturing process.

FIG. 3 shows an example of a pseudo random pattern of abrasive grainsconsistent with the present invention. A pseudo random pattern is anarrangement of abrasive grains that does not include a visuallyidentifiable pattern to a human observer. However, in a pseudo-randompattern, the same seed in an algorithm that determines placement of theabrasive grains will result in the same distribution or placement ofabrasive grains on an abrasive tool every time the algorithm isexecuted. While a pseudo-poisson arrangement of abrasive grains is usedas an example in the present application, other specific pseudo-randomabrasive arrangements will be apparent to one of skill in the art uponreading the present disclosure and are within the scope of the presentinvention.

Unlike a purely random pattern, a pseudo random pattern can provide theadvantage of a relatively even distribution of abrasive grains on anabrasive tool without the manufacturing constraints created by a regularpattern of abrasive grains. Further, random patterns can have relativelylarge working surface areas without abrasive grain coverage, resultingin inferior performance in that particular area, while other areas mayhave a high concentration of abrasive grains, resulting in excessive orvisual scratching resulting from use of that portion of the abrasivetool.

Unlike a regular pattern (as shown in FIG. 1), a pseudo random patterncan provide greater orientation independence, as defined according tothe Orientation Independence Test, so that it can be used to abrade orscour at any orientation with comparable results. Additionally,relatively minor manufacturing defects in abrasive grain placement arevisually apparent in the instance of a regular or patterned arrangementof abrasive grains. But a pseudo-random pattern consistent with thepresent application provides for more manufacturing flexibility.

In some instances, a pseudo-random pattern can comprise clusters ofabrasives grains, where each cluster includes a plurality of abrasivegrains arranged in a pseudo-random pattern, and where the clusters ofabrasive grains form an array on a substrate in a regular orpseudo-random arrangement.

Generally, a cluster is a subsection of the hand-pad containing at least3 features such that the percent of local area coverage of abrasivegrains in the cluster is greater than the percent total area coverage ofthe abrasive grains in the hand pad.

As shown in FIG. 3, the pseudo random pattern is a pseudo-poissonpattern with a 14% abrasive grain coverage. A pseudo-poisson pattern isa variation on a traditional poisson distribution and is described inmore detail in the Examples section. While 14% abrasive grain coverageis shown in FIG. 3, an abrasive tool consistent with the presentdisclosure may have a range of abrasive grain coverage, for example,abrasive grains may cover 10% to 30% of the working surface of anabrasive tool. Alternatively, an abrasive tool may have abrasive graincoverage in the range of 10% to 20% or 10% to 15%. While more abrasivegrain coverage on a tool may lead to faster or more efficient cutting,other factors such as freer cutting to allow debris to fall away easilyfrom the tool, cooler grinding or abrading, and manufacturing efficiencycan result from lower abrasive grain coverage.

Abrasive tool 30 includes a substrate 32 and abrasive grains 33. In FIG.3, the abrasive tool 30 is a hand pad that can be used for scouring orscrubbing; however, an abrasive tool consistent with the presentdisclosure can be any type of substrate with abrasive grains present.Examples of abrasive tools include tools for grinding, scouring,polishing, cutting, cleaning, finishing and sanding.

A substrate consistent with the present disclosure can include anysubstrate, such as paper, cloth, woven fabrics, nonwoven fabrics,calendared nonwoven fabrics, polymeric films, stitch-bonded fabrics,open cell foams, closed cell foams, vulcanized fiber materials, scrims,films, foils, screens, perforated sheets, other web-like substrates andcombinations thereof. A substrate can include a single material withdifferent types of treatments on different parts of the material. Forexample, a substrate made from a non-woven web may include asemi-densified layer as described by U.S. Patent Publication2017/0051442 to Endle et al., incorporated herein by reference.

An abrasive grain may refer to single abrasive grits, engineered,structured or shaped cutting points, abrasive agglomerates, or abrasiveparticles, composites comprising a plurality of abrasive grits orcomposites thereof. Examples of abrasive grits include diamond, cubicboron nitride, boron, suboxide, various alumina grains, such as fusedalumina, sintered alumina, seeded or unseeded sintered sol gel alumina,alumina zirconia grits, oxy-nitride alumina grits, silicon carbide,tungsten carbide, titanium carbide, garnet, iron oxide, tin oxide,feldspar, flint, emery, and modifications and combinations thereof. Suchabrasive grits may exhibit a Mohs hardness in the range of 8-10 Mohs.Other materials that exhibit sufficient hardness to provide a scouringfunction may include, for example, particles of melamine-formaldehyderesin, phenolic resin, polymethl methacrylate, polystyrene,polycarbonate, certain polyesters and polyamides, and the like. Suchmaterials may have a hardness in the range of at least 3 Mohs.

An abrasive grain may be any size consistent with the desiredapplication for the abrasive tool. A composite having a plurality ofabrasive grits in a resin may have a diameter in the range of 0.10 mm to5 mm. Or may diameter in a range defined by any of 0.10 mm, 0.50 mm, 1mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm, 4 mm or 5 mm. A composite may have aheight in the range of 0.05 mm to 2 mm. Or may have a height in a rangedefined by any of 0.10 mm, 0.25 mm, 0.50 mm, 0.75 mm, 1.0 mm, 1.5 mm or2 mm.

Examples of shaped abrasive particles can be found in U.S. Pat. No.5,201,916 (Berg); U.S. Pat. No. 5,366,523 (Rowenhorst RE 35,570); andU.S. Pat. No. 5,984,988 (Berg). U.S. Pat. No. 8,034,137 (Erickson etal.) describes alumina crushed abrasive particles that have been formedin a specific shape, then crushed to form shards that retain a portionof their original shape features. In some instances, shaped alphaalumina particles are precisely-shaped (i.e., the particles have shapesthat are at least partially determined by the shapes of cavities in aproduction tool used to make them.)

Structured or shaped abrasive particles may be desirable for moreefficient or higher precision grinding, polishing or abradingapplications. In these types of tools, small shaped compositestructures, such as three dimensional pyramids, diamonds, lines, andhexagonal ridges are replicated in a regular pattern on a surface of atool.

Abrasive agglomerates can include single abrasive particles bondedtogether with, for example, a polymer, a ceramic, a metal or a glass toform an abrasive agglomerate.

Composites comprising a plurality of abrasive grits can include any typeof abrasive grit discussed herein, or another type of abrasive grit thatwould be apparent to use to one skilled in the art upon reading thepresent disclosure. Abrasive grits can be combined with a range ofresins, such as thermosetting resins, UV curable resins, solvent-basedresins, including, in a non-limiting fashion, resins such as phenolicresins, aminoplast resins, curable acrylic resins, cyanate resins,urethanes, latex resins, nitrile resin, ethylene vinyl acetate resin,polyurethane resin, polyurea or urea-formaldehyde resin, isocyanateresin, styrene-butadiene resin, styrene-acrylic resin, vinyl acrylicresin, melamine resin, polyisoprene resin, epoxy resin, ethylenicallyunsaturated resin, and combinations thereof.

Abrasive grains may be aligned and placed on a substrate using analignment tool as described in U.S. Patent Publication No. 2016/037250.In another instance, direct transfer of abrasive grains onto thesubstrate may be carried out by placing a droplet of bonding material onthe substrate at the proper location and using a robotic arm to placeeach abrasive grain on the substrate. A robotic arm may also be used toplace a suspended array of abrasive on a substrate that is pre-coatedwith a bonding material. In some instances, a bonding material may be aresin as discussed herein. In some instances, a bonding material may bean adhesive.

In some instances, abrasive grains (and particularly grits in resin) maybe printed on an abrasive substrate using a screen printing or otherprinting method as will be apparent to one of skill in the art uponreading the present disclosure.

Abrasive grains may be bonded to a substrate using an adhesive makecoat, or they may be affixed directly to a substrate. Adhesives orbonding materials used to secure an abrasive grain to a substrate willdepend on the particular abrasive grain and substrate. Examples ofbonding materials include adhesives, brazing materials, electroplatingmaterials, electromagnetic materials, electrostatic materials, vitrifiedmaterials, metal powder bond materials, polymeric materials and resinmaterials and combinations thereof.

While the abrasive tool shown in FIG. 3 has a rectangular substrate andis designed for lateral movement, an abrasive tool may be a variety ofgeometric shapes and may be designed for rotation instead of lateralmovement. For example, an abrasive tool may have a shape such as arectangle, disk, rim, ring, cylinder, belt, conical, irregular shapes(such as in dental drills) or any combination of these shapes.

Variations on the present invention will be apparent to one of skill inthe art upon reading the present disclosure, and are within the scope ofthe invention set forth herein.

Examples

Example patterns for abrasive hand-pad articles were generated. Thescratch patterns on a simulated surface were analyzed. And the resultsare shown below.

Test Methods

Orientation Independence Test (OIT)

OIT Test Overview

The OIT analyzed the number of unique scratches made by Active Areas(AAs) chosen on a hand-pad (a type of abrasive tool). The active areaswere defined as sections of the hand-pad encompassing a subset of thefeatures used to abrade a substrate. The analysis consisted of randomlychosen (AAs) on the hand-pad surface that were of equal size andencompassed 10 to 50 abrasive features or abrasive grains. FIG. 4 showsan example of five circular AAs 41, 42, 43, 44, 45 chosen on a hand-pad40 with a hexagonally close-packed (Hex) abrasive pattern.

To conduct the OIT, linear scratches made or simulated with the chosenAAs 41, 42, 43, 44, 45 covering cutting directions ranging from 0 to 90degrees in one degree increments, where 0 degree cutting direction isdefined to be in the X direction were evaluated. FIG. 5 illustratescutting direction. Axis 51 shows the alignment of the abrasive grainswith X and Y coordinates. Diagram 52 shows a cut direction at a 0 degreeangle. Diagram 53 shows a cut direction at a 90 degree angle.

The OIT linear scratch test can be simulated or performed on a physicalabrasive tool. The simulated test assumes that both the 2D coordinatesof the centers of the abrasive features, and the average featurediameter on the hand-pad are known. Also note that the simulated testcan be performed from a physical hand-pad or other type of abrasive toolproviding these inputs are made available. For a step-by-step procedureon how the inputs can be obtained from a physical hand-pad, refer to thesection titled ‘OIT Test Input’ below.

OIT Test Input

For the simulated OIT test described, the tests take as input: 1) the 2Dcoordinate of the centers of the abrasive features, and 2) the averagediameter of the envelope that surrounds each abrasive feature. Themethod to obtain these two inputs for a physical hand pad or otherabrasive tool can be done through image analysis and is illustrated inthe steps shown in FIG. 6 and described below:

Step 1: Image Capture 61 of Abrasive Tool

Place abrasive tool onto a flat surface so that the abrasive grains (orabrasive features) are clearly visible. Arrange a camera so that thelens points in the direction approximately perpendicular to thehand-pad. Place the camera sufficiently far to capture the entiresurface of the abrasive tool in the camera's field of view and capturethe image. Abrasive tool 64 in FIG. 6 is an example of an image captureof a hand-pad with a hexagonal abrasive grain pattern. Also shown inimage 64 is the definition of the XY plane (that is, the XY plane isdefined as the plane spanned by the flat abrasive tool surface). Forabrasive tools that are not flat, a rectangular representative surfacearea of the abrasive tool can be extracted, flattened, and used for thefollowing analysis.

Step 2: Acquire Coordinates 62 of Abrasive Grain Centers

Define one corner of the hand-pad as (0, 0) and generate list of thecoordinates of the centers of the abrasive features. Image 65 in FIG. 6shows an example of acquiring the coordinates of three abrasive grains65 a (coordinates (84, 2.5)), 65 b (coordinates (72, 2.5)), and 65 c(coordinates (60, 2.5)) on the hand-pad.

Step 3: Acquire Average Diameter 63 of the Abrasive Grains

Define the average diameter of the abrasive grains or abrasive featuresas:

$D_{Ave} = {\frac{1}{N_{pad}}{\sum\limits_{i = 1}^{N_{pad}}D_{i}}}$

where D_(i) is the diameter of each abrasive grain. When an abrasivegrain is non-circular, D_(i) is measured as the diameter of the smallestcircle that can fit around the abrasive grain. N_(pad) is the number ofgrains on the abrasive pad or in the sampled section. Image 66 in FIG. 6shows an example of acquiring the abrasive grain diameter three of theabrasive grains 66 a, 66 b, 66 c on the abrasive tool. Each of theabrasive grains 66 a, 66 b, 66 c had a diameter of 2.5 mm as shown inImage 66.

Performing the OIT Test

The OIT method uses a scratch profile made by N_(OIT) cutting points foranalysis. N_(OIT) should be in the range of 10 to 50 cutting points.N_(OIT) represents the number of abrasive grains in an AA.

For each cutting direction the number of unique scratches found on theabraded substrate were counted. Multiple cutting points may result inscratches that are very close or even overlapping. Scratches that werecloser to each other than 10% of the average abrasive grain diameter(D_(AVE)) were counted as a single scratch. The analysis was repeatedfor the 90 cutting directions considered and normalize by the maximumnumber of unique scratches found. The OIT score is the differencebetween the maximum and minimum number of unique scratches found in the90 cutting directions considered. The pass/fail criteria is defined tobe patterns that exhibit OIT scores of less than 0.20 (i.e.,OIT<0.20=pass). FIG. 7 shows an example of the output 70 of a simulatedOIT for the abrasive tool shown in FIG. 4. Points 73, 74 and 75 are allareas where multiple abrasive features scratched along the same line.Orientation Independence is measured by the difference between thehighest and lowest point on the output 70. Difference 72 isapproximately 0.20 for the abrasive tool shown in FIG. 4, and thus not apassing score on the OIT.

Local Homogeneity Test (LHT)

The LHT evaluates the homogeneity of the entire abrasive tool bycalculating the Homogeneity Index (HI) each of multiple sections locatedon the abrasive tool. Each section considered was chosen such N_(LHT) ofthe nearest abrasive grains were included, and N_(LHT) was in the rangeof 40 to 70. When performing the LHT on the abrasive tool shown in FIG.4, the abrasive tool was divided up into sections that were evenlyspaced 1 mm apart. Each section comprised a region containing 50 of thenearest abrasive grains. Each section was analyzed according to equationbelow. The LHT analyzes each section and creates a heat map based on theHI score of each section. The HI score for a section on the abrasivetool is given by:

${{HI} = {1 - \frac{\frac{1}{N_{LHT}}{\sum\limits_{i = 1}^{N_{LHT}}\left( {d_{i} - d_{theory}} \right)^{2}}}{d_{theory}}}},{d_{theory} = \sqrt{\frac{2A}{N_{LHT}\sqrt{3}}}}$

Where:

N_(LHT) is the number of features in the bounding box;

A is the area of the bounding section; and

d_(i) is the nearest neighbor distance of the i-th feature in thebounding section.

The resulting HI is associated with the abrasive grain at the center ofa given section

The pass criteria for the LHT is defined as all sections on an abrasivetool having an HI score between 0.7 and 0.9 (i.e., (0.7<HI<0.90)=pass).

FIG. 8 shows a heat map of the output of a LHT for the abrasive grainpattern shown in FIG. 5. In the heat map shown in FIG. 8, the LHI scoreis: (min, max)=(0.9395, 0.9956) (a failing score).

Empirical observations of LHT results of several abrasive tool examplesindicated that abrasive tools with regions having HI<0.70 was found toexhibit poor average feature spacing control. On the other hand, regionswith HI>0.90 are regions of high symmetry which would manifest itself inpoor OIT performance implying sensitivity of the hand-pad to varyingcutting directions. Passing both tests (OIT and LHT) according to thepassing criteria described herein ensures that the patterns exhibit 1)consistent feature to feature spacing across the hand-pad, and 2) aninsensitivity to cutting direction, regardless of which section of theabrasive tool is used.

Simulated Abrasive Grain Examples

A total of eight different simulated abrasive grain arrangements weretested under the OIT and the LHT. The eight abrasive grain arrangementsincluded four different “patterns”, with each pattern being tested attwo different levels of abrasive grain coverage, 14% and 26%,respectively.

Hex Pattern:

Two of the patterns were abrasive grains arranged in a regular hexpattern. The 14% coverage hex pattern is shown in FIG. 10a (referred toas Comparative Example 1 or CE 1), and the 26% coverage hex pattern isshown in FIG. 10b (referred to as Comparative Example 2 or CE 2). FIG. 9shows the basic arrangement of abrasive grains use to create the hexpattern in FIGS. 10a and 10b . The hex pattern shown in FIG. 9 was twointerpenetrating rectangular grids where the second grid was offset by cand d in the horizontal and vertical directions respectively. Distancesa, b, c, and d as labeled in FIG. 9 are shown in the table below for CE1and CE2.

TABLE 1 Quantities to generate the hex comparative example patterns Hexat 14% (CE1) Hex at 26% (CE2) a 11.0 mm 8.5 mm b 6.5 mm 4.5 mm c 5.5 mm4.25 mm d 3.25 mm 2.25 mm

Vogel Pattern:

Two of the patterns were abrasive grains arranged in a Vogel pattern.The 14% coverage Vogel pattern is shown in FIG. 10c (referred to asComparative Example 3 or CE 3), and the 26% coverage Vogel pattern isshown in FIG. 10d (referred to as Comparative Example 4 or CE 4). TheVogel pattern was defined by the following equation:

r=c√{square root over (n)},θ=ng

Where n represents a positive integer and indexes the number of abrasivegrains generated, c represents a positive real number, and g representsan irrational number approximately equal to 2.39996 radians (anapproximation of the golden angle). In the present work, c=1 mm and sothe (x,y) coordinate of the n-th feature in the pattern was given by:

(X,Y)_(n)=√{square root over (n)}(cos(ng), sin(ng))

Pseudo-Random Pattern:

Two of the patterns were abrasive grains arranged in a Pseudo-Random(Pseudo-Poisson). The 14% coverage pseudo-random pattern is shown inFIG. 10e (referred to as Example 1 or E1). The 26% coveragepseudo-random pattern is shown in FIG. 10f (referred to as Example 2 orE2). The pseudo code below explains how the locations for each point inthe pseudo-random pattern of Example 1 and Example 2 were generated.

CODE FOR PSEUDO-POISSON PATTERN GENERATION Inputs: PadX; Width ofabrasive pad PadY; Height of abrasive pad x ₀ ; Initial position of1^(st) dot. Bold quantities denote 2D vectors. R_(ave); Average radiusof dots Coverage; Area coverage of dots α; User-prescribed to controlspacing variance (Δx, Δy); Vector of maximum allowable deviation fromminimum. Prescribed to control randomness Start of pseudo-code PadArea =PadX * PadY; area of rectangular abrasive pad DotArea = 3.14159 *Rave{circumflex over ( )}2; area of each dot N_(pts) = Floor[(Coverage *Pad Area)/(DotArea)]; Number of points for desired coverage X ₀ = x ₀ ;1st point (taken as input) d = Sqrt[ PadX{circumflex over ( )}2 +PadY{circumflex over ( )}2]; distance from periodic images of 1^(st)point Energy = α / d{circumflex over ( )}2 Energy increase due toinsertion of 1^(st) point For i=1 to N_(pts) Loop to fill abrasive padarea with points X = Minimize[Energy + alpha / (X _(i−1) − X) · Find Xthat minimizes energy. (X _(i−1) − X)]; Minimize[ ] is a standardtechnique in numerical methods. The function here searches for X in theabrasive pad that minimizes the value in the parenthesis [Energy0 +alpha / (X _(i−1) − X) · (X _(i−1) − X). ‘·’ denotes dot product betweentwo vectors Xi = X + [Rand([−1, 1]) * Δx; Perturbed position to insertRand[−1, 1]) * Δy]; randomness. Rand[(−1, 1)] returns random numberbetween −1 to 1. Energy = Energy + alpha / (X _(i−1) − X _(i)) · (X_(i−1) − X _(i)); endfor end of pseudo-code Outputs: X ₀ , X ₁ , .... ,X _(Npts) Positions of all the points in the pattern

Random Pattern:

Two of the patterns were abrasive grains arranged in a random manner.The 14% coverage random pattern is shown in FIG. 10g (referred to asComparative Example 5 or CE5). The 26% coverage random pattern is shownin FIG. 10h (referred to as Comparative Example 5 or CE5). The randomarrangement of particles was generated using Mathematica version 10.3 (acommercial mathematical analysis tool available form Wolfram, Champaign,Ill.) random number generator, a random hand-pad pattern was generatedby prescribing the minimum distance between each feature, and generatingenough points so as to cover the entire hand-pad at the prescribedcoverage.

In each of the Examples/Comparative Examples, the feature diameters areall 2.5 mm, and the XY coordinates are can be derived as describedherein. The OIT and LHT described above were performed on each of theExamples/Comparative Examples described. The results from these testsare summarized in Table 2.

Results

TABLE 2 Results from the OIT and LHT of the 8 different abrasive featurepatterns % Area LHI: Cover- OIT OIT: LHI LHI Pass/ Example age ScorePass/Fail Min Max Fail E1 14 0.0571 Pass 0.7528 0.8801 Pass E2 26 0.0554Pass 0.7410 0.8780 Pass CE1 14 0.2000 Fail 0.9395 0.9956 Fail CE2 260.1510 Pass 0.8751 0.9673 Fail CE3 14 0.0667 Pass 0.7799 0.9166 Fail CE426 0.0585 Pass 0.7722 0.9398 Fail CE5 14 0.0476 Pass 0.5461 0.7039 FailCE6 26 0.0037 Pass 0.6419 0.7891 Fail

Of the various patterns tested, only E1 and E2 exhibited passing scoresfor both the Orientation Independent Test and the Local HomogeneityTest.

1. An abrasive tool comprising a substrate and plurality of abrasivegrains arranged in a pseudo-random pattern on the substrate, wherein theabrasive grains cover in the range of 10% to 30% of the substratesurface and wherein the arrangement of abrasive grains has a score of20% or less on the Orientation Independence Test and has a score in therange of 0.7 to 0.9 in the Local Homogeneity Index Test.
 2. The abrasivetool of claim 1, wherein the arrangement of abrasive grains has a scoreof 10% or less on the Orientation Independence Test.
 3. The abrasivetool of claim 1, wherein the abrasive grains cover 10% to 15% of thesurface substrate.
 4. The abrasive tool of claim 1, wherein thesubstrate comprises a lofty non-woven material.
 5. The abrasive tool ofclaim 4, wherein the lofty non-woven material comprises a densifiedsurface.
 6. The abrasive tool of claim 1, wherein the abrasive grainsare at least one of: single abrasive grits, cutting points, andcomposites comprising a plurality of abrasive grits, and combinationsthereof.
 7. The abrasive tool of claim 6 wherein the composites comprisea plurality of abrasive grits in a resin.
 8. The abrasive tool of claim1, wherein the abrasive grains are printed onto the substrate.
 9. Theabrasive tool of claim 1, wherein the average abrasive grain height fromthe surface of the substrate is in the range of 0.25 mm to 1.5 mm. 10.The abrasive tool of claim 1, wherein the pseudo-random patterncomprises clusters of abrasive grains.
 11. The abrasive tool of claim 1,wherein the substrate is selected from the group consisting of paper,woven fabrics, nonwoven fabrics, calendared nonwoven fabrics, polymericfilms, stitchbonded fabrics, open cell foams, closed cell foams, andcombinations thereof.
 12. The abrasive tool of claim 1, wherein thesubstrate comprises an open cell foam or a closed cell foam laminated toa substrate selected from the group consisting of paper, woven fabrics,nonwoven fabrics, calendared nonwoven fabrics, polymeric films,stitchbonded fabrics, open cell foams, closed cell foams, andcombinations thereof.
 13. The abrasive tool of claim 1, wherein thepseudo-random pattern is a pseudo-poisson pattern. 14-24. (canceled)